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With Interactive Exploration of a Dynamical System earlier this month and today’s Scrubbing Calculator, Bret Victor is doing some of the most provocative work in math education right now. As much as I’m curious how STEM educators perceive his work, I admit I’m much more interested in the perceptions of educators in the humanities and of people who, at one point or another, were totally put off by the abstract symbology of mathematics.

If our goal is to empower students to do more and more interesting mathematics, we can’t just hand them simulators and tell them to go play: we need to teach them how to create those simulators. Doing that requires a lot of math and a lot of programming. So Victor’s “simulation” model of doing math ultimately requires teaching kids a lot of traditional mathematics.

Of course, replace “simulators” with “calculators” and we have another familiar argument. Obviously, I’m of two very different minds about this.

18 Comments

LSquared

I agree with you. The simulators are cool, but you don’t learn the same stuff from them that you do from math. If this is our strategy, we might become better at teaching our students to split costs of a trip, or calculate a tip (assuming that they have an iPhone), and students might even understand what predator prey equations do. What we don’t get is students who can design an iPhone app, or invent a better predator-prey differential equation. And we do need at least some of those students.

Tim

Math is comprised of many things; ideas, pictures, equations, logic. I have said many times myself that I must grind through some “awful calculation” in order to obtain the result that I’m trying to prove. I think it’s important to communicate generally to others. Once motivation and context have been developed, you can try to find the best model and then ask: What is mathematically true? Can you prove/justify it? Inevitably, calculations and logic are necessary to make such advances. If the techniques already exist then by all means, farm it out to technology for quicker results. If not then well, you’ve got some new mathematics to create. This is at the heart of how mathematics is done and I believe we do students a disservice by hiding this inherent messiness.

Evan

I studied mechanical engineering as an undergraduate before going into teaching. My tolerance for abstract math in the absence of any engineering application was fairly low – it wasn’t until I started teaching math that I really got excited about the beauty of math for its own sake. This has resulted in major internal struggle (and neither ever ends up winning) between the practical and theoretical (or skills-based) activities we do in my classroom.

I think the simulation is nice in that it enables students to be active in their explorations of a topic. While a simulation may not improve understanding of the underlying mathematics, it can enable students to have a bit of intuition about how changing the parameters of the simulation affects its behavior. That intuition can then be confirmed if students go back and study the theoretical side and develop their mathematical background.

On the other hand, if all you do is teach the skills and background, you’ll lose those students that seemingly have a giant “what’s-the-point” light flashing in their vision every time they step into your classroom. Without the background though, the only way students can figure out a problem is through trial and error – a very long process for practical problems that are better solved using more analytical methods.

So often this type of conversation is framed as doing one or the other – even the internal conversations I have with myself about what to do in class on a particular day. I’ve had to force myself to actively find ways to eat the curriculum from both ends – background then application, then application then background, etc. At the end of the day, I feel better about switching back and forth as a teacher, and I think the students appreciate (and benefit from) seeing both approaches.

The thing about differential equations is that you don’t actually need to know any formal calculus to understand what they mean, and “simulating” them (i.e. solving them numerically) requires essentially no mathematics beyond algebra.

In fact, there is a pretty wonderful calculus textbook, freely available online, that begins with the formulation and simulation of a system of three differential equations to model the spread of a disease. It’s here:

There’s something really mathematical in the Scrubbing Calculator, which is that the slider takes the place of a variable.

The technique of taking several guesses at the value of a variable in order to judge what is happening is pretty great, and big in Common Core’s standards for mathematical practice (it’s #8).

But then the algebra vanishes, since the slider can be used to just directly calculate the “right” value of the variable. What I’d rather see is a slider that you can use for a while, then its value vanishes and becomes “b” for bar height.

I want students to understand that the calculations and operations they perform on numbers are the same as the ones they perform on variables. A lot of kids miss this connection between arithmetic and algebra. The slider and the work of the Scrubbing Calculator could do a lot to reinforce these connections.

Kill Math, in my opinion, goes too far in trying to remove algebra altogether from the process — they even say this: “most people are not comfortable with bundling up meaning into abstract symbols and making them dance. Thus, the power of math beyond arithmetic is generally reserved for a clergy of scientists and engineers”. I think these tools can be great if they lead to conversations about the utility and meaning of variable, but it doesn’t look like that’s the intent.

The biggest thing worrying me about this treatment of variable (or rather, non-treatment) is that everything kind of looks and feels the same after a while. Students wouldn’t get the feel of linear versus quadratic versus exponential, or really much sense of variation at all.

It’s a step up from some other places I’ve seen sliders used in math courses, but could really go much further, and be used in conjunction with symbolic mathematics, rather than attempting to replace it.

Bowen: The biggest thing worrying me about this treatment of variable (or rather, non-treatment) is that everything kind of looks and feels the same after a while. Students wouldn’t get the feel of linear versus quadratic versus exponential, or really much sense of variation at all.

This is a strong point. Not a problem that couldn’t be overcome (with, say, the visualization tool he assigns us for homework) but it’d be a shame to lose one’s intuitive sense of quadratic v. exponential v. linear growth.

First, one can design a simulation in two ways. Either the simulation produces correct results, which means you get at least one solution to the given problem if such a solution exists. Or there is something wrong with the underlying code so that the simulation is not consistent with the given problem and produces incorrect results or no results at all.

For the student, there is – apart from a heuristic approach – no way to check whether the simulation correctly implements the given rules and correctly acts on the variables. As Dan pointed out, this brings us back to the discussion on graphing calculators and how they act as a ‘black box’.

What we see in Bret Victor’s video is undoubtedly a very nice and clean, interactive alternative to a MATLAB output, but playing around with the time slider and the phase space plot to again demonstrate the correctness of the implementation must not be anything else than the last step in dealing with the problem.

The second problem is that every minute of class that is spent on teaching how to operate calculators or on how to produce MATLAB code is a minute that is lost for the actual math.

Glad to see this conversation developing. A friend alerted me to the Kill Math project just 2 weeks ago, and I had mixed reactions as well –

Pro) Anybody looking to create more entry points into mathematical content is okay by me. In a way, Bret Victor’s ideas here feel continuous to me with Tristan Needham’s fantastic book Visual Complex Analysis, which is basically a big reference on complex analysis in which as much as possible, all arguments are geometric and illustrated by diagrams. Nathan Carter tried the same thing for group theory. (I don’t experience Carter’s book as quite as much of a revelation, but it’s still pretty fabulous, and it might be different for somebody else.) In all three cases the idea is – there are a lot of extremely conceptually rich pictures implicit in the content here; let’s show them to people. (In Victor’s case, there’s continuous motion in addition to still images.) This is always going to be a good thing, and since imagining and rendering helpful images is significant mathematical and pedagogical work, I am glad people are doing it. I have complete confidence that the images and products of Victor’s project are going to be awesome for math and math education.

(He writes that education is explicitly not his concern. He wants to change math. Or replace math. Or something. Of course Carter and Needham are not trying to replace anything, just add to it. Viewed as additions, to either math or math education, the stuff is awesome in all these cases.)

Con) Something about the tone really bugged me. I guess I’ve frequently enough been guilty of thinking that I KNOW WHAT THE PROBLEM IS so I KNOW HOW TO FIX IT, but he seems to be thinking that way and it is irking me. He has his eye on one part of the picture. (Re: math, math education, both.) Only one part. There are a lot of other parts that he hasn’t thought deeply about. This is true of all of us. I guess from the outside it’s easy to see that it makes overzealousness unbecoming. (Note to self.)

Something about the command line analogy bugs me in particular, but I don’t have time to get into it now.

Natalie S

While I find the scrubbing calculator incredibly interesting and potentially very valuable, I, too, have many questions.

First, in looking at Victor’s examples, the inherent question to me is, how did he set up the “equations” in the first place? Understanding that the total height in pixels needed to be 768, and that the height of the bars could vary to fill the given space indicates an understanding of a “variable”. While my introductory algebra students may not immediately use an x to indicate the variable, they do understand that it is changing. They also understand “guess and check” methods for solving such a problem. But isn’t the beauty of algebra that we don’t have to guess and check, but that using variables will yield an exact answer in very few steps?

Additionally, if students have the ability to set up an equation that uses a “scrubbing” type of solution strategy, aren’t they already quite developed in their understanding of Algebra, or in algebraic thinking, if not solving problems algebraically?
Or does the “scrubbing calculator” come across as more “mathmagic” to students in introductory Algebra?

Without understanding properties such as order of operations, students are once again facing the challenge of trusting a calculator without knowing for sure if the solution is reasonable. Sure, anyone can say, look at the equation, the numbers add up to 768! But that means we have to believe the equation was set up correctly in the first place and that the program we used to “scrub” followed the steps correctly. Too often we trust answers on calculators when they don’t see the nuances in what we intended, assumed parentheses, the difference between a negative and subtraction, etc. Using a scrubbing tool all students will arrive at the same answer, and you lose the rich mathematical discussion on: 1. How different people could arrive at different answers. 2. How to solve problems in a variety of ways and 3. What strategies are mathematically valid and why they are so.

Looking at his “double scrubble” example of the car trip problem, understanding that 2910-1000 and 426 +1000 needed to be equal seems like a fairly sophisticated problem solving solution for this situation. Again, students with this understanding could probably solve the 3-step equation he started with, and would have set it up themselves anyway. I suspect, though I can’t be sure, that most of my students would have added the two amounts together, divided by two and then found the difference. Oh, yes, that’s what the calculator does for you, without the same understanding of what is happening. Seems to me that his method of “removing the symbols” is still more sophisticated than the basic computation that most of us would use. If we are going to advocate for simplicity, let’s recognize that many times using Algebra is NOT necessary for routine problem solving situations.

While I am all about making math accessible to all, I do feel that both of the problems Victor presents are ones that all of us should be able to solve by implementing a strategy slightly more advanced than guess and check (often times without a variable). While we don’t want students to get so bogged down in details that they lose both interest and confidence in their ability to do math, can’t we advocate for a balance between mathematical understanding and real-world practicality? Why does it have to be one or the other?

As always, Dan, you raise such thought-provoking questions, thank you for the opportunity to reflect on what is best for our students.

I would argue that replacing “simulations” with “calculators” in his quote isn’t the same thing. Calculators are a general-purpose tool that replace manual algorithms with digital ones. Bret’s examples have all been incredibly problem-specific, not general-purpose at all, and as such don’t replace general-purpose descriptors.

Also, I dunno, isn’t almost everything he’s talking about already being done? Is he even aware of the Wolfram Demonstrations site?

I love graphing and visual representations, I will always use them as part of my teaching wherever I can, and more actual tools to create those visualizations would be fantastic. But I don’t see how his ideas actually solve the problems he’s talking about.

re: reaction of humanities folks, I can try poking a couple I know to see what they say, but my go-to for a couple years on the disconnect between the mathematical and the nonmathematical has been Adam Cadre’s review of The Pleasures of Counting.

Don’t use glyphs when you can use words. But I guess then he wouldn’t be talking to mathematicians, who doubtless don’t have to unpack a sigma the way I do, and would instead be talking to me, which he doesn’t want to do.

I have some sympathy for Bret here in that he’s worried about “symbolic overload” which has glazed the eyes of many a student. I’ve lately trying to use design to alleviate the problem, whereas he wants to do away with symbols altogether. I’m worried his solution doesn’t work in general (for reasons other people have already brought up). I’m especially concerned about second-level variation — like having students experiment try to match an exponential decay equation ax^b with a particular set of data, you have one kind of variable with the x but also varying the parameters a and b, so you’ve got one level of scrubbing with the x but an entirely different type of scrubbing for the a and b.

I guess Bret’s solution to that would be more like his dynamic exploration demonstration, but how would students set the problem up in the first place WITHOUT referring to at least x?

You have to give me specifics so I can understand, and then once we’re on the same page, then we can talk about the abstract case! No abstractions till I can picture it in my head, and I’ve never seen a county with n towns in it. Talk about, I dunno, eight towns instead. And for heaven’s sake, if we’re going to be talking about a flow of 3 trains or 4 trains per hour, do not label the towns 3 and 4 and so on! That’s confusing! Give the towns some fricking names. Amsterdam, Berlin, Copenhagen.

One issue that I am struggling with, especially as I look at Bret Victor’s stuff, is the way to connect “his world” (the “math”-without-doing-“math”-world) and “our world” (the belief in abstraction as a good). Teachers who believe in the power of abstract thought, and in the importance of teaching abstract ideas, will naturally turn themselves off to Victor’s proposals.

But I wonder if we’re not discussing a point here that is tangential to the real issue. I have no doubt that Victor is quite capable of “doing the math” that we would ask our own students to do, whatever that means. However, I suspect that some of his ideas would be really positive ways to get _younger_ students into mathematics and general quantitative thinking quicker and more fluidly. Many of the comments I’m seeing here are focusing on high school-age students, implicitly or otherwise, and perhaps that’s not where the power of his ideas lies.

Which would do more damage to 5-9 year olds, in terms of their ultimate attitude toward and appreciation of the power of quantitative thought: giving them worksheet after worksheet after timed test, or letting them work on real problems, and teaching them how to use something like the “Scrubber” that Victor proposes to figure out real answers? I don’t really care that there aren’t lots of people who can write simulations and program calculators – what I do care about is the poor attitude of most towards math. I’d be willing to bet that an improvement in attitudes would be joined by an improvement in understanding.

Like Victor and most of you I assume, I have or had the “gift” to understand math from childhood and finally acquiring a master degree in it. Like most of you I’ve taught math, albeit only for three years (I’m working in software engineering now), and I’ve done extensive tutoring with students who did not have the gift.

Like Victor I have had a growing uneasiness with math being such an important subject for a vast majority of people not being interested in it, and like him I felt the problem was with math and not with the students. Leaving a general tendency of sloth in the Western world out of the equation, I too believe math should be replaced in the curriculum with other subjects (programming, experimental data analysis, formal philosophy, music) which lead to similar objectives (logic, scientific language, culture, beauty) AND with a more immediate appeal, not only for coolness sake, but for applicability to the real world.

Most contras worry about obscuring math for those who now specialize in it, like us. I believe this is false, much like having driving lessons on the curriculum, instead of mechanical engineering, has not led to shortage of mechanical engineers making cars. Math specialists will still find their way to it, when intrigued by the mechanics under the hood of applications like Victor’s. It’s those other 99% which never grow into a math career but still find it on their curriculum, who would benefit of killing math in favour of a more intuitive scientific language, in particular a more visual one.

I think we are watching a genius at work and should not treat his visions lightly.